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compass and straightedge construction

In order to define what a compass and straightedge construction is, some preliminary definitions are necessary:

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A compass is a tool which can be used for drawing circles, or arcs thereof, whose radii are sufficiently small, and for measuring lengths.

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A straightedge is a tool which can be used for drawing lines, or segments thereof.

Some people use the word ruler to refer to the tool that is called a straightedge here. This can cause some confusion, however, because outside of mathematics, a ruler is used to measure any length desired. This is not a permissible tool for compass and straightedge constructions.

With these preliminaries out of the way, we can now proceed to the main definition.

A compass and straightedge construction is the provable creation of a geometric figure on the Euclidean plane (or complex plane) such that the figure is created using only a compass, a straightedge, and specified geometric figures.

Typically, if no preexisting geometric figure is specified, the tacit assumption is that one can use a line segment of length 11. Moreover, in such instances, one can specify what length represents 11, but it must remain constant throughout the construction.

A geometric figure is constructible if it can be made from a compass and straightedge construction.

One has to be very careful with the terminology associated with compass and straightedge constructions. For example, the phrase “A 20∘superscript20 angle is not constructible with compass and straightedge” refers to the fact that a compass and straightedge construction of a 20∘superscript20 angle is not possible using the tacit assumption as described above. As another example, the following is an erroneous argument regarding compass and straightedge constructions:

A line segment of length π𝜋 is constructible because, given a line segment of length 11, I can extend it as a ray. Then I can measure the distance between the two endpoints with my compass. After that, I can open the compass π𝜋 times wider. Finally, I can mark that distance on the ray from one of the endpoints of the original line segment.

The above argument is one of many reasons why the word provable appears in the definition. One can open the compass however wide one wants, and one can mark arcs and points however one wants, but the construction is invalid unless one can prove indisputably that the construction really is what is stated. In the above example, one cannot prove that the compass was opened exactlyπ𝜋 times wider than the length of the original line segment.

Compass and straightedge constructions are of historical significance. The ancient Greeks are the most well-known civilization for investigating these constructions on an elementary level. It should be pointed out that the compasses that they used were collapsible. That is, you could open the compass and draw an arc, but immediately after you removed a point of the compass from the plane where you drew the arc, the compass would close completely. It turns out that whether a collapsible compass or a modern-day compass is used to perform these constructions makes no difference. This statement is justified by the fact that one can use a collapsible compass to construct a circle with a given radius at any point as shown by this entry.

One of the greatest applications of abstract algebra is being able to determine which constructions are possible and which are not. The connection between constructions and abstract algebra is that the set of all constructible points is in one-to-one correspondence with the elements of the field of constructible numbers. Without abstract algebra, one would be hard pressed to prove statements about constructibility such as “A 20∘superscript20 angle is not constructible with compass and straightedge.”

Mathematics Subject Classification

Comments

This is an attempt to reduce the standard axiom set for group.
Let S be a set with an associative law defined everywhere: for any a and b in S, there exists an unique c such that ab = c.
This associative law satisfies the following axioms:

1 - There exists at least one right-neutral element e (may be more f, g...) such that ae = a for all a in S (af=a, ag=a...)

2 - Every a in S has at least one right inverse a' (maybe more) with respect to one of the right elements e, such that aa' = e.

Axiom 2 differs from the standard definition: for a and b in S, there are right inverse a' and b', not necessarily unique, such that aa' = e and bb' = f, but e and f could be different right-neutral elements.

1. Under the proposed axiom set, there exists at most one right inverse, but there could exist none, for you only say the right-hand side of $ab=c$ exists. If I ask you to construct $x^{-1}$ for some $x$ in the group, what $a$ and $b$ do you take?

What if the set is the integers with multiplication?
For any integer $a$ and $b$ there certainly is another unique integer $c$ such that $ab=c$. e.g. $2 \times 3 = 6$, but $2 \times 3 \neq 4$.

2. Along the same lines, what is the procedure for constructing the identity $e$? Well, you might say $x x^{-1} = e$, but this is of course circular with (1).

> This associative law satisfies the following axioms:
>
> 1 - There exists at least one right-neutral element e (may
> be more f, g...) such that ae = a for all a in S (af=a,
> ag=a...)
>
> 2 - Every a in S has at least one right inverse a' (maybe
> more) with respect to one of the right elements e, such that
> aa' = e.
>

Do you mean there is a "right-neutral element" e such that, for each a in S, there is a' in S such that aa'=e?

If "minimizing" the group axiom set means "minimizing" the number of identities required to define a "group", then it has been shown that a group is equivalent to an algebraic system whose underlying set is the underlying set of the group, with one binary operation called "division" and one identity.

No, these two articles deal with a method of writing a given set of axioms in a more compact form. What I am presenting is a different set of axioms. Maybe the term "miminizing" is not appropriate, I meant in fact, a weaker set of axioms.

Standard set of axioms for the law of a group G:
1 - The law is associative.
2 - There exists a neutral element e sucha that ae = ea = a for all a in G.
3 - All a in G have an inverse a' such that aa' = a'a = e.

"Weak" set of axioms:
1 - The law is associative.
2 - There exists a right-neutral element e such that ae = a for all a in G.
3 - All a in G have a right inverse a' such that aa' = e.

It can be proven that the weak set is equivalent to the "stong" set by considering the product aa'a", where a" is the right inverse of a'. Then, we get the following results:
I - a right inverse is also a left inverse.
II - the right-neutral element is also left-neutral
III - the right inverse is unique, as is the neutral element.

What I suggest is to try to make axiom 3 even weaker: the inverses of a', b' of a and b are such that aa' = e and bb' = f, and the righ-elements e and f could be different. If we try to apply the former method, we get a problem:
a' being the right inverse of a, aa' = e. If a" is the right inverse of a', a'a" = f and f could be different from e. From the product aa'a" we get the following results:
I - a'a = f. The right inverse is also left inverse but with respect to another right-neutral element.
II - ea = a. The right-neutral element a is also left-neutral for a only. if bb' = f for example, then f will be left-neutral for b and not for a.

The question is: is there a way to complete the proof and get the standard set of axioms?